Optimality in the development of intestinal crypts

Intestinal crypts display a developmental temporal order in which the establishment of stem cells precedes the expansion of nonstem cells. Optimal control theory predicts that this strategy minimizes the time needed to create a mature crypt.

The mouse small intestine is a classic model system for mammalian stem cell biology. Intestinal crypts in adults comprise a minority of long-lived stem cells, which continuously divide to maintain their numbers while producing shorter lived differentiated nonstem cells. The availability of intestinal stem cell markers and of mouse models that enable tracking the fates of individual stem cell divisions makes this an ideal system to study stem cell dynamics during the complex processes of tissue development, homeostasis and repair.

Figure: Bang-bang control of intestinal crypt development. A – Expanding crypts minimize the time to achieve a mature crypt by first creating stem cells (red) through symmetric divisions and only later switching to asymmetric divisions, generating nonstem cells (blue) with a delay. B – Resulting dynamics of stem cells and nonstem cells predicted from the bang-bang optimal control. C – Fluorescence detection of Lgr5 transcripts (green) in developing intestinal crypts. D – Short term lineage tracing of progenies of Lgr5 stem cells (labeled in red) reveal a switch from symmetric stem cell divisions to asymmetric divisions.

In this work we asked what are the design principles that govern the dynamic proportions of stem cells and nonstem cells in developing crypts. Intestinal crypts appear shortly after birth as small indentations and are rapidly expanded towards their mature size. The crypt expansion process is governed by the way in which stem cells divide. Symmetric stem cell divisions expand the stem cell pool, whereas asymmetric divisions create new nonstem cells. Both the dynamic crypt composition and the total time required to expand the crypt to its mature size depend on the way in which stem cells dynamically modulate these modes of divisions. But what is the stem cell proliferation strategy that will attain the mature crypt at the minimal time?

To address this question we devised a mathematical model of crypt growth in which stem cells dynamically allocate their division resources according to a stem cell control function p(t). At any time along the developmental process a fraction p(t) of the existing stem cells divide symmetrically to make more stem cells whereas a second fraction, 1-p(t), divide asymmetrically to generate new nonstem cells. To understand which control function leads to a mature crypt at the minimal time we used the mathematical tools of optimal control theory, a theory that is commonly used in economics and engineering to solve resource allocation problems. Using this theory we found that the mature crypt would be attained in the minimal time if stem cells employ a division strategy known as a ‘bang-bang’ control. In this strategy the crypt expansion process consists of two distinct phases – in the first phase all stem cells should divide symmetrically, rapidly expanding the mature crypt stem cell pool while delaying production of nonstem cells (p(t)=1). In the second phase all stem cells should sharply transition to asymmetric divisions (p(t)=0), maintaining stem cell numbers while generating the nonstem cells with a delay.

To test our theoretical predictions we used single molecule fluorescence in-situ hybridization of the Lgr5 stem cell marker in infant mice to count the numbers of stem cells in crypts of increasing sizes, at successive stages along the crypt developmental process. We found that small crypts are at first entirely made up of stem cells. Above a critical size, crypts continue to grow by adding Lgr5-negative nonstem cells. We also tracked the divisions of individual Lgr5 stem cells using short-term lineage tracing and found that a predominance of stem cell symmetric divisions precedes a switch to asymmetric divisions, confirming our theoretical predictions. Our approach, combining theoretical models with sensitive detection of individual cells in intact tissues, can be used to analyze complex processes of development, homeostasis and repair in other tissues.